An efficient numerical scheme to solve generalized Abel’s integral equations with delay arguments utilizing locally supported RBFs

Abstract

Hereditary effects are commonly observed in diverse scientific domains such as engineering, economics, biology, mathematics, and physics. In the model of atomic irradiation of solids with unbounded cross-sectional areas, determining the average number of atoms displaced has been achieved through delay systems that incorporate the consideration of past states. In this study, we employ the discrete collocation method using local radial basis functions to numerically solve Abel-type integral equations with a delay argument. This method balances accuracy, efficiency, and flexibility and is well-suited for complex practical problems as it requires less memory and computational volume compared to its global counterpart. Instead of approximating the solution at all points of the domain, the method considers a set of nodes in the neighborhood of a certain point. As a result, the method can be readily implemented on a standard personal computer without requiring high-end specifications. To calculate the singular integrals the nonuniform composite Gauss–Legendre numerical integration rule is employed. We discuss the error analysis and convergence rate of the offered scheme and test it with several numerical examples. The results obtained are also consistent with the theoretical error analysis expectations.